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Journal articles on the topic 'Unstable periodic orbits (UPOs)'

Author: Grafiati

Published: 22 July 2024

Last updated: 31 July 2025

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1

Saiki, Y., and M. Yamada. "Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems." Nonlinear Processes in Geophysics 15, no. 4 (2008): 675–80. http://dx.doi.org/10.5194/npg-15-675-2008.

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Abstract. Unstable periodic orbit (UPO) recently has become a keyword in analyzing complex phenomena in geophysical fluid dynamics and space physics. In this paper, sets of UPOs in low dimensional maps are theoretically or systematically found, and time averaged properties along UPOs are studied, in relation to those of chaotic orbits.

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COY, BENJAMIN. "DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING." International Journal of Bifurcation and Chaos 22, no. 01 (2012): 1230001. http://dx.doi.org/10.1142/s0218127412300017.

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An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of t

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Morena, Matthew A., and Kevin M. Short. "Cupolets: History, Theory, and Applications." Dynamics 4, no. 2 (2024): 394–424. http://dx.doi.org/10.3390/dynamics4020022.

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In chaos control, one usually seeks to stabilize the unstable periodic orbits (UPOs) that densely inhabit the attractors of many chaotic dynamical systems. These orbits collectively play a significant role in determining the dynamics and properties of chaotic systems and are said to form the skeleton of the associated attractors. While UPOs are insightful tools for analysis, they are naturally unstable and, as such, are difficult to find and computationally expensive to stabilize. An alternative to using UPOs is to approximate them using cupolets. Cupolets, a name derived from chaotic, unstabl

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Dolan, Kevin, Annette Witt, Jürgen Kurths, and Frank Moss. "Spatiotemporal Distributions of Unstable Periodic Orbits in Noisy Coupled Chaotic Systems." International Journal of Bifurcation and Chaos 13, no. 09 (2003): 2673–80. http://dx.doi.org/10.1142/s021812740300817x.

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Techniques for detecting encounters with unstable periodic orbits (UPOs) have been very successful in the analysis of noisy, experimental time series. We present here a technique for applying the topological recurrence method of UPO detection to spatially extended systems. This approach is tested on a network of diffusively coupled chaotic Rössler systems, with both symmetric and asymmetric coupling schemes. We demonstrate how to extract encounters with UPOs from such data, and present a preliminary method for analyzing the results and extracting dynamical information from the data, based on a

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TIAN, YU-PING, and XINGHUO YU. "STABILIZING UNSTABLE PERIODIC ORBITS OF CHAOTIC SYSTEMS WITH UNKNOWN PARAMETERS." International Journal of Bifurcation and Chaos 10, no. 03 (2000): 611–20. http://dx.doi.org/10.1142/s0218127400000426.

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A novel adaptive time-delayed control method is proposed for stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems with unknown parameters. We first explore the inherent properties of chaotic systems and use the system state and time-delayed system state to form an asymptotically stable invariant manifold so that when the system state enters the manifold and stays in it thereafter, the resulting motion enables the stabilization of the desired UPOs. We then use the model following concept to construct an identifier for the estimation of the uncertain system parameters. We shal

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Boukabou, A., A. Chebbah, and A. Belmahboul. "Stabilizing Unstable Periodic Orbits of the Multi-Scroll Chua's Attractor." Nonlinear Analysis: Modelling and Control 12, no. 4 (2007): 469–77. http://dx.doi.org/10.15388/na.2007.12.4.14678.

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This paper addresses the control of the n-scroll Chua’s circuit. It will be shown how chaotic systems with multiple unstable periodic orbits (UPOs) detected in the Poincar´e section can be stabilized as well as taking the system dynamics from one UPO to another.

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Maiocchi, Chiara Cecilia, Valerio Lucarini, and Andrey Gritsun. "Decomposing the dynamics of the Lorenz 1963 model using unstable periodic orbits: Averages, transitions, and quasi-invariant sets." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 3 (2022): 033129. http://dx.doi.org/10.1063/5.0067673.

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Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters’ value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics’ period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approxim

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8

Saiki, Y. "Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors." Nonlinear Processes in Geophysics 14, no. 5 (2007): 615–20. http://dx.doi.org/10.5194/npg-14-615-2007.

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Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

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Gritsun, A. "Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1991 (2013): 20120336. http://dx.doi.org/10.1098/rsta.2012.0336.

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The theory of chaotic dynamical systems gives many tools that can be used in climate studies. The widely used ones are the Lyapunov exponents, the Kolmogorov entropy and the attractor dimension characterizing global quantities of a system. Another potentially useful tool from dynamical system theory arises from the fact that the local analysis of a system probability distribution function (PDF) can be accomplished by using a procedure that involves an expansion in terms of unstable periodic orbits (UPOs). The system measure (or its statistical characteristics) is approximated as a weighted sum

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10

TIAN, YU-PING. "AN OPTIMIZATION APPROACH TO LOCATING AND STABILIZING UNSTABLE PERIODIC ORBITS OF CHAOTIC SYSTEMS." International Journal of Bifurcation and Chaos 12, no. 05 (2002): 1163–72. http://dx.doi.org/10.1142/s0218127402005017.

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In this paper, a novel method for locating and stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems is proposed. The main idea of the method is to formulate the UPO locating problem as an optimization issue by using some inherent properties of UPOs of chaotic systems. The global optimal solution of this problem yields the desired UPO. To avoid a local optimal solution, the state of the controlled chaotic system is absorbed into the initial condition of the optimization problem. The ergodicity of chaotic dynamics guarantees that the optimization process does not stay forever

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11

HINO, TORU, SHIGERU YAMAMOTO, and TOSHIMITSU USHIO. "STABILIZATION OF UNSTABLE PERIODIC ORBITS OF CHAOTIC DISCRETE-TIME SYSTEMS USING PREDICTION-BASED FEEDBACK CONTROL." International Journal of Bifurcation and Chaos 12, no. 02 (2002): 439–46. http://dx.doi.org/10.1142/s0218127402004450.

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In this paper, we consider feedback control that stabilizes unstable periodic orbits (UPOs) of chaotic discrete-time systems. First, we show that there exists a strong necessary condition for stabilization of the UPOs when we use delayed feedback control (DFC) that is known as one of the useful methods for controlling chaotic systems. The condition is similar to that in the fixed point stabilization problem, in which it is impossible to stabilize the target unstable fixed point if the Jacobian matrix of the linearized system around it has an odd number of real eigenvalues greater than unity. I

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Wang, Zhen, Yong Xin Li, Xiao Jian Xi, and Xian Feng Wang. "Computional Dynamics for Diffusionless Lorenz Equations with Periodic Parametric Perturbation." Advanced Materials Research 905 (April 2014): 651–54. http://dx.doi.org/10.4028/www.scientific.net/amr.905.651.

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The dynamics of diffusionless Lorenz equations (DLE) with periodic parametric perturbation is studied through numerical and experimental investigations in this paper. A method for calculating Lyapunov exponents (LEs), Lyapunov dimension (LD) from time series is presented. Furthermore, bifurcation and some complex dynamic behaviors such as periodic, quasi-periodic motion and chaos which occurred in the system are analyzed. And an algorithm for detecting unstable periodic orbits (UPOs) is presented. Also, give some numerical simulation studies of the system in order to verify the analytic result

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UETA, TETSUSHI, GUANRONG CHEN, and TOHRU KAWABE. "A SIMPLE APPROACH TO CALCULATION AND CONTROL OF UNSTABLE PERIODIC ORBITS IN CHAOTIC PIECEWISE-LINEAR SYSTEMS." International Journal of Bifurcation and Chaos 11, no. 01 (2001): 215–24. http://dx.doi.org/10.1142/s0218127401002092.

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This paper describes a simple method for calculating unstable periodic orbits (UPOs) and their control in piecewise-linear autonomous systems. The algorithm can be used to obtain any desired UPO embedded in a chaotic attractor, and the UPO can be stabilized by a simple state feedback control. A brief stability analysis of the controlled system is also given.

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14

Moroz, Irene M. "Template analysis of a nonlinear dynamo." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2137 (2011): 288–302. http://dx.doi.org/10.1098/rspa.2011.0216.

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In this paper, we extend our previous template analysis of a self-exciting Faraday disc dynamo with a linear series motor to the case of a nonlinear series motor. This introduces two additional nonlinear symmetry-breaking terms into the governing dynamo equations. We investigate the consequences for the identification of a possible template on which the unstable periodic orbits (UPOs) lie. By computing Gauss linking numbers between pairs of UPOs, we show that their values are not incompatible with those for a template for the Lorenz attractor for its classic parameter values.

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DELEANU, DUMITRU. "STABILIZING THE PERIODIC ORBITS IN A CHAOTIC MAPPING DESCRIBING THE DISCRETE HEALTH SYSTEMS VIA PREDICTION-BASED CONTROL." Journal of marine Technology and Environment 2021, no. 2 (2021): 21–26. http://dx.doi.org/10.53464/jmte.02.2021.04.

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In the paper the problem of location and stabilization of unstable periodic orbits (UPOs) in discrete systems is investigated via the prediction-based control (PBC). It involves using the state of the free system one period ahead as reference for the control signal. Two types of control gains are tested, the first requiring the knowledge of the UPO to be stabilized and the second depending only on the actual state of the trajectory. The effectiveness of PBC is demonstrated on a chaotic mapping describing the malignant tumor growth. When the results obtained with the two control laws are compar

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Deleanu, Dumitru. "Detecting and stabilizing periodic orbits of chaotic Henon map through predictive control." Annals Constanta Maritime University 27, no. 2018 (2018): 73–78. http://dx.doi.org/10.38130/cmu.2067.100/42/12.

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The predictive control method is one of the proposed techniques based on the location and stabilization of the unstable periodic orbits (UPOs) embedded in the strange attractor of a nonlinear mapping. It assumes the addition of a small control term to the uncontrolled state of the discrete system. This term depends on the predictive state ps + 1 and p(s + 1) + 1 iterations forward, where s is the length of the UPO, and p is a large enough nonnegative integer. In this paper, extensive numerical simulations on the Henon map are carried out to confirm the ability of the predictive control to dete

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17

Lee, Byoung-Cheon, Ki-Hak Lee, and Bo-Hyeun Wang. "Control Bifurcation Structure of Return Map Control in Chua's Circuit." International Journal of Bifurcation and Chaos 07, no. 04 (1997): 903–9. http://dx.doi.org/10.1142/s0218127497000704.

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We demonstrate that return map control and adaptive tracking can be used together to locate, stabilize, and track unstable periodic orbits (UPOs). Through bifurcation studies as a function of some control parameters of return map control, we observe the control bifurcation (CB) phenomenon which exhibits another route to chaos. Nearby an UPO there are a lot of driven periodic orbits (DPOs) along the CB route. DPOs are not embedded in the original chaotic attractor, but they are generated artificially by driving the system slightly in a direction with feedback control. Based on the CB phenomenon

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18

Zhu, Qunxi, Xin Li, and Wei Lin. "Leveraging neural differential equations and adaptive delayed feedback to detect unstable periodic orbits based on irregularly sampled time series." Chaos: An Interdisciplinary Journal of Nonlinear Science 33, no. 3 (2023): 031101. http://dx.doi.org/10.1063/5.0143839.

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Detecting unstable periodic orbits (UPOs) based solely on time series is an essential data-driven problem, attracting a great deal of attention and arousing numerous efforts, in nonlinear sciences. Previous efforts and their developed algorithms, though falling into a category of model-free methodology, dealt with the time series mostly with a regular sampling rate. Here, we develop a data-driven and model-free framework for detecting UPOs in chaotic systems using the irregularly sampled time series. This framework articulates the neural differential equations (NDEs), a recently developed and

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19

AKATSU, SATOSHI, HIROYUKI TORIKAI, and TOSHIMICHI SAITO. "ZERO-CROSS INSTANTANEOUS STATE SETTING FOR CONTROL OF A BIFURCATING H-BRIDGE INVERTER." International Journal of Bifurcation and Chaos 17, no. 10 (2007): 3571–75. http://dx.doi.org/10.1142/s021812740701938x.

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This paper studies stabilization of low-period unstable periodic orbits (UPOs) in a simplified model of a current mode H-bridge inverter. The switching of the inverter is controlled by pulse-width modulation signal depending on the sampled inductor current. The inverter can exhibit rich nonlinear phenomena including period doubling bifurcation and chaos. Our control method is realized by instantaneous opening of inductor at a zero-crossing moment of an objective UPO and can stabilize the UPO instantaneously as far as the UPO crosses zero in principle. Typical system operations can be confirmed

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20

LEKEBUSCH, A., A. FÖRSTER, and F. W. SCHNEIDER. "CHAOS CONTROL BY ELECTRIC CURRENT IN AN ENZYMATIC REACTION." International Journal of Neural Systems 07, no. 04 (1996): 393–97. http://dx.doi.org/10.1142/s0129065796000361.

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We apply the continuous delayed feedback method of Pyragas to control chaos in the enzymatic Peroxidase-Oxidase (PO) reaction, using the electric current as the control parameter. At each data point in the time series, a time delayed feedback function applies a small amplitude perturbation to inert platinum electrodes, which causes redox processes on the surface of the electrodes. These perturbations are calculated as the difference between the previous (time delayed) signal and the actual signal. Unstable periodic P1, 11, and 12 orbits (UPOs) were stabilized in the CSTR (continuous stirred ta

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Oteski, L., Y. Duguet, and L. R. Pastur. "Lagrangian chaos in confined two-dimensional oscillatory convection." Journal of Fluid Mechanics 759 (October 27, 2014): 489–519. http://dx.doi.org/10.1017/jfm.2014.583.

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AbstractThe chaotic advection of passive tracers in a two-dimensional confined convection flow is addressed numerically near the onset of the oscillatory regime. We investigate here a differentially heated cavity with aspect ratio 2 and Prandtl number 0.71 for Rayleigh numbers around the first Hopf bifurcation. A scattering approach reveals different zones depending on whether the statistics of return times exhibit exponential or algebraic decay. Melnikov functions are computed and predict the appearance of the main mixing regions via the break-up of the homoclinic and heteroclinic orbits. The

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Kazantsev, E. "Sensitivity of the attractor of the barotropic ocean model to external influences: approach by unstable periodic orbits." Nonlinear Processes in Geophysics 8, no. 4/5 (2001): 281–300. http://dx.doi.org/10.5194/npg-8-281-2001.

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Abstract. A description of a deterministic chaotic system in terms of unstable periodic orbits (UPO) is used to develop a method of an a priori estimate of the sensitivity of statistical averages of the solution to small external influences. This method allows us to determine the forcing perturbation which maximizes the norm of the perturbation of a statistical moment of the solution on the attractor. The method was applied to the barotropic ocean model in order to determine the perturbation of the wind field which provides the greatest perturbation of the model's climate. The estimates of per

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23

Miino, Yuu, Daisuke Ito, Tetsushi Ueta, and Hiroshi Kawakami. "Locating and Stabilizing Unstable Periodic Orbits Embedded in the Horseshoe Map." International Journal of Bifurcation and Chaos 31, no. 04 (2021): 2150110. http://dx.doi.org/10.1142/s0218127421501108.

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Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate locat

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Ivan, Cosmin, and Mihai Catalin Arva. "Nonlinear Time Series Analysis in Unstable Periodic Orbits Identification-Control Methods of Nonlinear Systems." Electronics 11, no. 6 (2022): 947. http://dx.doi.org/10.3390/electronics11060947.

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The main purpose of this paper is to present a solution to the well-known problems generated by classical control methods through the analysis of nonlinear time series. Among the problems analyzed, for which an explanation has been sought for a long time, we list the significant reduction in control power and the identification of unstable periodic orbits (UPOs) in chaotic time series. To accurately identify the type of behavior of complex systems, a new solution is presented that involves a method of two-dimensional representation specific to the graphical point of view, and in particular the

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Katsanikas, M., Víctor J. García-Garrido, and S. Wiggins. "Detection of Dynamical Matching in a Caldera Hamiltonian System Using Lagrangian Descriptors." International Journal of Bifurcation and Chaos 30, no. 09 (2020): 2030026. http://dx.doi.org/10.1142/s0218127420300268.

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The goal of this paper is to apply the method of Lagrangian descriptors to reveal the phase space mechanism by which a Caldera-type potential energy surface (PES) exhibits the dynamical matching phenomenon. Using this technique, we can easily establish that the nonexistence of dynamical matching is a consequence of heteroclinic connections between the unstable manifolds of the unstable periodic orbits (UPOs) of the upper index-1 saddles (entrance channels to the Caldera) and the stable manifolds of the family of UPOs of the central minimum of the Caldera, resulting in the temporary trapping of

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Lucarini, Valerio, and Andrey Gritsun. "A new mathematical framework for atmospheric blocking events." Climate Dynamics 54, no. 1-2 (2019): 575–98. http://dx.doi.org/10.1007/s00382-019-05018-2.

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Abstract We use a simple yet Earth-like hemispheric atmospheric model to propose a new framework for the mathematical properties of blocking events. Using finite-time Lyapunov exponents, we show that the occurrence of blockings is associated with conditions featuring anomalously high instability. Longer-lived blockings are very rare and have typically higher instability. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking event, while a relative increase of predictability is found in the mature phase. The opposite holds for Pacific blo

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27

Keeler, Jack S., Alice B. Thompson, Grégoire Lemoult, Anne Juel, and Andrew L. Hazel. "The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2232 (2019): 20190434. http://dx.doi.org/10.1098/rspa.2019.0434.

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We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within a Hele-Shaw channel containing a depth-perturbation. Recent experiments revealed that the bubble shape becomes more complex, quantified by an increasing number of transient bubble tips, with increasing flow rate. Eventually, the bubble changes topology, breaking into multiple distinct entities with non-trivial dynamics. We demonstrate that qualitatively simi

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EL AROUDI, A., M. DEBBAT, R. GIRAL, G. OLIVAR, L. BENADERO, and E. TORIBIO. "BIFURCATIONS IN DC–DC SWITCHING CONVERTERS: REVIEW OF METHODS AND APPLICATIONS." International Journal of Bifurcation and Chaos 15, no. 05 (2005): 1549–78. http://dx.doi.org/10.1142/s0218127405012946.

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This paper presents, in a tutorial manner, nonlinear phenomena such as bifurcations and chaotic behavior in DC–DC switching converters. Our purpose is to present the different modeling approaches, the main results found in the last years and some possible practical applications. A comparison of the different models is given and their accuracy in predicting nonlinear behavior is discussed. A general Poincaré map is considered to model any multiple configuration of DC–DC switching converters and its Jacobian matrix is derived for stability analysis. More emphasis is done in the discrete-time app

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Dong, Chengwei, and Lian Jia. "Periodic orbits analysis for the Zhou system via variational approach." Modern Physics Letters B 33, no. 19 (2019): 1950212. http://dx.doi.org/10.1142/s0217984919502129.

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We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipativ

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30

Eccles, F. J. R., P. L. Read, and T. W. N. Haine. "Synchronization and chaos control in a periodically forced quasi-geostrophic two-layer model of baroclinic instability." Nonlinear Processes in Geophysics 13, no. 1 (2006): 23–39. http://dx.doi.org/10.5194/npg-13-23-2006.

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Abstract. Cyclic forcing on many timescales is believed to have a significant effect on various quasi-periodic, geophysical phenomena such as El Niño, the Quasi-Biennial Oscillation, and glacial cycles. This variability has been investigated by numerous previous workers, in models ranging from simple energy balance constructions to full general circulation models. We present a numerical study in which periodic forcing is applied to a highly idealised, two-layer, quasi-geostrophic model on a β-plane. The bifurcation structure and (unforced) behaviour of this particular model has been extensivel

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Savi, Marcelo A., Francisco Heitor I. Pereira-Pinto, and Armando M. Ferreira. "Chaos Control in Mechanical Systems." Shock and Vibration 13, no. 4-5 (2006): 301–14. http://dx.doi.org/10.1155/2006/545842.

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Chaos has an intrinsically richness related to its structure and, because of that, there are benefits for a natural system of adopting chaotic regimes with their wide range of potential behaviors. Under this condition, the system may quickly react to some new situation, changing conditions and their response. Therefore, chaos and many regulatory mechanisms control the dynamics of living systems, conferring a great flexibility to the system. Inspired by nature, the idea that chaotic behavior may be controlled by small perturbations of some physical parameter is making this kind of behavior to b

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Guha, Anirban, and Firdaus E. Udwadia. "Nonlinear dynamics induced by linear wave interactions in multilayered flows." Journal of Fluid Mechanics 816 (March 6, 2017): 412–27. http://dx.doi.org/10.1017/jfm.2017.84.

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Using simple kinematics, we propose a general theory of linear wave interactions between the interfacial waves of a two-dimensional (2D), inviscid, multilayered fluid system. The strength of our formalism is that one does not have to specify the physics of the waves in advance. Wave interactions may lead to instabilities, which may or may not be of the familiar ‘normal-mode’ type. Contrary to intuition, the underlying dynamical system describing linear wave interactions is found to be nonlinear. Specifically, a saw-tooth jet profile with three interfaces possessing kinematic and geometric symm

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33

So, Paul. "Unstable periodic orbits." Scholarpedia 2, no. 2 (2007): 1353. http://dx.doi.org/10.4249/scholarpedia.1353.

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Carpintero, D. D., and J. C. Muzzio. "The Lyapunov exponents and the neighbourhood of periodic orbits." Monthly Notices of the Royal Astronomical Society 495, no. 2 (2020): 1608–12. http://dx.doi.org/10.1093/mnras/staa1227.

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ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to

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35

Chizhevsky, V. N., and P. Glorieux. "Targeting unstable periodic orbits." Physical Review E 51, no. 4 (1995): R2701—R2704. http://dx.doi.org/10.1103/physreve.51.r2701.

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KATSANIKAS, M., P. A. PATSIS, and G. CONTOPOULOS. "INSTABILITIES AND STICKINESS IN A 3D ROTATING GALACTIC POTENTIAL." International Journal of Bifurcation and Chaos 23, no. 02 (2013): 1330005. http://dx.doi.org/10.1142/s021812741330005x.

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We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four-dimensional spaces of section, we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stabilit

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Pawelzik, K., and H. G. Schuster. "Unstable periodic orbits and prediction." Physical Review A 43, no. 4 (1991): 1808–12. http://dx.doi.org/10.1103/physreva.43.1808.

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Katsanikas, Matthaios, and Stephen Wiggins. "Phase Space Structure and Transport in a Caldera Potential Energy Surface." International Journal of Bifurcation and Chaos 28, no. 13 (2018): 1830042. http://dx.doi.org/10.1142/s0218127418300422.

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We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of

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KATSANIKAS, M., P. A. PATSIS, and G. CONTOPOULOS. "THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION." International Journal of Bifurcation and Chaos 21, no. 08 (2011): 2321–30. http://dx.doi.org/10.1142/s0218127411029811.

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We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the correspondin

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Parker, Jeremy P., and Tobias M. Schneider. "Invariant tori in dissipative hyperchaos." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 11 (2022): 113102. http://dx.doi.org/10.1063/5.0119642.

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One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system’s chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems, including fluid turbulence, while higher-dimensional invariant tori representing quasiperiodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperch

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KAMIYAMA, KYOHEI, MOTOMASA KOMURO, and TETSURO ENDO. "BIFURCATION OF QUASI-PERIODIC OSCILLATIONS IN MUTUALLY COUPLED HARD-TYPE OSCILLATORS: DEMONSTRATION OF UNSTABLE QUASI-PERIODIC ORBITS." International Journal of Bifurcation and Chaos 22, no. 06 (2012): 1230022. http://dx.doi.org/10.1142/s0218127412300224.

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In this paper, we obtain bifurcations of quasi-periodic orbits occuring in mutually coupled hard-type oscillators by using our recently developed computer algorithm to directly determine the unstable quasi-periodic orbits. So far, there is no computer algorithm to draw unstable invariant closed curves on a Poincare map representing quasi-periodic orbits. Recently, we developed a new algorithm to draw unstable invariant closed curves by using the bisection method. The results of this new algorithm are compared with the previously obtained averaging method results. Several new results are found,

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Zhang, Yongxiang, and Guanwei Luo. "Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems." International Journal of Non-Linear Mechanics 96 (November 2017): 12–21. http://dx.doi.org/10.1016/j.ijnonlinmec.2017.07.011.

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Bradley, Elizabeth, and Ricardo Mantilla. "Recurrence plots and unstable periodic orbits." Chaos: An Interdisciplinary Journal of Nonlinear Science 12, no. 3 (2002): 596–600. http://dx.doi.org/10.1063/1.1488255.

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Voros, A. "Unstable periodic orbits and semiclassical quantisation." Journal of Physics A: Mathematical and General 21, no. 3 (1988): 685–92. http://dx.doi.org/10.1088/0305-4470/21/3/023.

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Pisarchik, A. N. "Dynamical tracking of unstable periodic orbits." Physics Letters A 242, no. 3 (1998): 152–62. http://dx.doi.org/10.1016/s0375-9601(98)00210-2.

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Dana, Itzhack. "Hamiltonian transport on unstable periodic orbits." Physica D: Nonlinear Phenomena 39, no. 2-3 (1989): 205–30. http://dx.doi.org/10.1016/0167-2789(89)90005-5.

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Fazendeiro, L., B. M. Boghosian, P. V. Coveney, and J. Lätt. "Unstable periodic orbits in weak turbulence." Journal of Computational Science 1, no. 1 (2010): 13–23. http://dx.doi.org/10.1016/j.jocs.2010.03.004.

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DHAMALA, MUKESHWAR, and YING-CHENG LAI. "THE NATURAL MEASURE OF NONATTRACTING CHAOTIC SETS AND ITS REPRESENTATION BY UNSTABLE PERIODIC ORBITS." International Journal of Bifurcation and Chaos 12, no. 12 (2002): 2991–3005. http://dx.doi.org/10.1142/s0218127402006308.

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The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable per

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OGORZAŁEK, MACIEJ J., and ZBIGNIEW GALIAS. "CHARACTERISATION OF CHAOS IN CHUA'S OSCILLATOR IN TERMS OF UNSTABLE PERIODIC ORBITS." Journal of Circuits, Systems and Computers 03, no. 02 (1993): 411–29. http://dx.doi.org/10.1142/s0218126693000253.

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We present a picture book of unstable periodic orbits embedded in typical chaotic attractors found in the canonical Chua's circuit. These include spiral Chua's, double-scroll Chua's and double hook attractors. The "skeleton" of unstable periodic orbits is specific for the considered attractor and provides an invariant characterisation of its geometry.

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Kazantsev, E. "Unstable periodic orbits and attractor of the barotropic ocean model." Nonlinear Processes in Geophysics 5, no. 4 (1998): 193–208. http://dx.doi.org/10.5194/npg-5-193-1998.

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Abstract. A numerical method for detection of unstable periodic orbits on attractors of nonlinear models is proposed. The method requires similar techniques to data assimilation. This fact facilitates its implementation for geophysical models. This method was used to find numerically several low-period orbits for the barotropic ocean model in a square. Some numerical particularities of application of this method are discussed. Knowledge of periodic orbits of the model helps to explain some of these features like bimodality of probability density functions (PDF) of principal parameters. These P

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